How can I lay down an object on a plane?
One solution is to use Blender's rigid body physics simulation. Basically, you simulate the object falling and settling on the floor from gravity.

In the Properties > Physics panel, turn on both Collision and Rigid Body for both your floor object and the object you want to lay down.



For the floor object, change the rigid body Type to Passive.



Press AltA to animate the physics simulation. You should see object fall down to the floor and eventually come to rest. Press AltA again to stop the animation. If you want to apply the final position to all frames, select the object and press CtrlA>Visual Transform, then turn off the physics for those objects in the Physics panel.